By Loring W. Tu

Manifolds, the higher-dimensional analogues of gentle curves and surfaces, are basic gadgets in smooth arithmetic. Combining facets of algebra, topology, and research, manifolds have additionally been utilized to classical mechanics, common relativity, and quantum box thought. during this streamlined advent to the topic, the idea of manifolds is gifted with the purpose of supporting the reader in attaining a quick mastery of the fundamental issues. via the tip of the booklet the reader could be in a position to compute, at the least for easy areas, some of the most easy topological invariants of a manifold, its de Rham cohomology. alongside the best way the reader acquires the data and abilities invaluable for extra research of geometry and topology. the second one version includes fifty pages of recent fabric. Many passages were rewritten, proofs simplified, and new examples and workouts additional. This paintings can be utilized as a textbook for a one-semester graduate or complex undergraduate path, in addition to through scholars engaged in self-study. The considered necessary point-set topology is integrated in an appendix of twenty-five pages; different appendices overview evidence from actual research and linear algebra. tricks and recommendations are supplied to some of the workouts and difficulties. Requiring in simple terms minimum undergraduate must haves, "An creation to Manifolds" can also be a good starting place for the author's booklet with Raoul Bott, "Differential varieties in Algebraic Topology."

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Wedge product and scalars Let V be a vector space. For a, b ∈ R, f ∈ Ak (V ), and g ∈ Aℓ (V ), show that a f ∧bg = (ab) f ∧g. 7. Transformation rule for a wedge product of covectors Suppose two sets of covectors on a vector space V , β 1 , . . , β k and γ 1 , . . , γ k , are related by βi = k ∑ aij γ j , i = 1, . . , k, j=1 for a k × k matrix A = [aij ]. Show that β 1 ∧ · · · ∧ β k = (det A) γ 1 ∧ · · · ∧ γ k . 8. Transformation rule for k-covectors Let f be a k-covector on a vector space V .

Show that α 1 ∧ · · · ∧ α k = 0 if and only if α 1 , . . , α k are linearly independent in the dual space V ∨ . * Exterior multiplication Let α be a nonzero 1-covector and γ a k-covector on a finite-dimensional vector space V . Show that α ∧ γ = 0 if and only if γ = α ∧ β for some (k − 1)-covector β on V . 34 §4 Differential Forms on Rn §4 Differential Forms on Rn Just as a vector field assigns a tangent vector to each point of an open subset U of Rn , so dually a differential k-form assigns a k-covector on the tangent space to each point of U.

Propositions A and B express the property d 2 = 0 of the exterior derivative on open subsets of R3 ; these are easy computations. Proposition C expresses the fact that a 1-form on R3 is exact if and only if it is closed. Proposition C need not be true on a region other than R3 , as the following well-known example from calculus shows. Example. If U = R3 − {z-axis}, and F is the vector field −y x , ,0 x2 + y2 x2 + y2 F= on R3 , then curlF = 0, but F is not the gradient of any C∞ function on U. The reason is that if F were the gradient of a C∞ function f on U, then by the fundamental theorem for line integrals, the line integral C − y x2 + y2 dx + x x2 + y2 dy over any closed curve C would be zero.